Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $q \neq 0$. $r = \dfrac{40q + 90}{q} \div \dfrac{16q + 36}{5q} $
Dividing by an expression is the same as multiplying by its inverse. $r = \dfrac{40q + 90}{q} \times \dfrac{5q}{16q + 36} $ When multiplying fractions, we multiply the numerators and the denominators. $r = \dfrac{ (40q + 90) \times 5q } { q \times (16q + 36) } $ $ r = \dfrac {5q \times 10(4q + 9)} {q \times 4(4q + 9)} $ $ r = \dfrac{50q(4q + 9)}{4q(4q + 9)} $ We can cancel the $4q + 9$ so long as $4q + 9 \neq 0$ Therefore $q \neq -\dfrac{9}{4}$ $r = \dfrac{50q \cancel{(4q + 9})}{4q \cancel{(4q + 9)}} = \dfrac{50q}{4q} = \dfrac{25}{2} $